Banach Journal of Mathematical Analysis

G-convergence and homogenization of monotone damped hyperbolic equations

Gabriel Nguetseng, Hubert Nnang, and Nils Svanstedt

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Abstract

Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form ${\frac{\partial^2 u^\omega_{\varepsilon}}{\partial t^2}} - \mathrm{div}\left(a\left(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_{\varepsilon}\right)\right)-\Delta(\frac{\partial u^\omega_{\varepsilon}}{\partial t}) +G\left(T_3(\frac{x}{\varepsilon_3})\omega_3 ,t,\frac{\partial u^\omega_{\varepsilon}}{\partial t}\right)=f$. It is shown, under certain structure assumptions on the random maps $a\left(\omega_1,\omega_2,t,\xi\right)$ and $G\left(\omega_3,t,\eta\right)$, that the sequence $\{u^\omega_\epsilon\}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem ${\frac{\partial^2 u}{\partial t^2}} - \mathrm{div}\left( b \left( t,D u \right)\right)-\Delta(\frac{\partial u}{\partial t})+{\overline G}(t,\frac{\partial u}{\partial t}) = f$.

Article information

Source
Banach J. Math. Anal., Volume 4, Number 1 (2010), 100-115.

Dates
First available in Project Euclid: 27 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1272374674

Digital Object Identifier
doi:10.15352/bjma/1272374674

Mathematical Reviews number (MathSciNet)
MR2593909

Zentralblatt MATH identifier
1192.35016

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 35B40: Asymptotic behavior of solutions

Keywords
G-convergence multiscale stochastic homogenization elliptic parabolic hyperbolic

Citation

Nguetseng, Gabriel; Nnang, Hubert; Svanstedt, Nils. G-convergence and homogenization of monotone damped hyperbolic equations. Banach J. Math. Anal. 4 (2010), no. 1, 100--115. doi:10.15352/bjma/1272374674. https://projecteuclid.org/euclid.bjma/1272374674


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